Abstract
The line-inversion and pedal transformation are defined in the quasi-hyperbolic plane and certain properties of these transformations are shown with regard to analogous transformations in the Euclidean [1, 3, 10, 12, 20], hyperbolic [4, 15, 18] isotropic [17, 19] and pseudo-Euclidean plane [5, 6, 7, 14]. As it is natural to observe class curves in the quasi-hyperbolic plane, i.e. line envelopes, the construction of a tangent point on any line of the class curve obtained by the line-inversion and pedal transformation is shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sohn (New York, 1969).
H. Halas, N. Kovačević and A. Sliepčević, Line-inversion in the quasi-hyperbolic plane, in: The 16th ICGG Proc. (Innsbruck, 2014), pp. 739–748.
Hirst T. A.: On the quadric inversion of the plane curves, Proc. Roy. Soc. London 14, 91–106 (1865)
Horváth Á. G.: Hyperbolic plane geometry revised, J. Geom. 106, 341–362 (2015)
M. Katić Žlepalo and E. Jurkin, Circular cubics and quartics obtained as pedal curves of conics in pseudo-Euclidean plane, in: The 15th ICGG Proc. (Montreal, 2012), pp. 341–347.
Kovačević N., Jurkin E.: Circular cubics and quartics in pseudo-Euclidean plane obtained by inversion, Math. Pannon. 22, 1–20 (2011)
Kovačević N., Szirovicza V.: Inversion in Minkowskischer Geometrie, Math. Pannon. 21, 89–113 (2010)
Makarova N. M.: On the projective metrics in plane, Učenye zap. Mos. Gos. Ped. in-ta 243, 274–290 (1965) (in Russian)
Milojević M. D.: Certain Comparative examinations of plane geometries according to Cayley–Klein, Novi Sad J. Math. 29, 159–167 (1999)
Niče V.: Curves and surfaces of the 3rd and 4th order deduced by quadratic inversion. Rad HAZU 278, 153–194 (1945) (in Croatian)
H. Sachs, Ebene Isotrope Geometrie, Friedr. Vieweg & Sohn (Braunschweig/Wiesbaden, 1987).
S. Salmon, Higher Plane Curves, Chelsea Publishing Company (New York, 1879).
Sliepčević A., Božic I., Halas H.: Introduction to the Planimetry of the Quasi-Hyperbolic Plane. KoG 17, 58–64 (2013)
Sliepčević A., Katić M. Žlepalo.: Pedal curves of conics in pseudo-Euclidean plane. Math. Pannon. 23, 75–84 (2012)
Sliepčević A., Szirovicza V.: A classification and construction of entirely circular cubics in the hyperbolic plane. Acta Math. Hungar. 104, 185–201 (2004)
Sommerville D. M. Y.: Classification of geometries with projective metric. Proc. Ediburgh Math. Soc. 28, 25–41 (1910)
Szirovicza V.: Die Fusspunktskurven der Kegelschnitten der isotropen Ebene. KoG 1, 3–5 (1996)
Szirovicza V.: Vollkommen zirkuläre Kurven Fusspunktkurven der hyperbolische Ebene. Rad JAZU 408, 17–25 (1984)
Szirovicza V., Sliepčević A.: Die allgemeine Inversion in der isotropen Ebene. Rad HAZU 491, 153–168 (2005)
H. Wieleitner, Spezielle Ebene Kurven, G. J. Göschen (Leipzig, 1908).
Yaglom I. M., Rozenfeld B. A., Yasinskaya E. U.: Projective metrics. Russ. Math. Surreys 19, 51–113 (1964)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Halas, H. Line-inversion and pedal transformation in the quasi-hyperbolic plane. Acta Math. Hungar. 151, 462–481 (2017). https://doi.org/10.1007/s10474-016-0686-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-016-0686-y